Singularly perturbed filters for dynamic average consensus

Solmaz S. Kia Jorge Cortes Sonia Martinez

Abstract— This paper proposes two continuous-time dynamicaverage consensus filters for networks described by balancedand weakly-connected directed topologies. Our distributedfilters, termed 1st-Order-Input ( FOI-DCF ) and 2nd-Order-Input Dynamic (SOI-DCF ) Consensus Filters, respectively,allow agents to track the average of their dynamic inputswithin an O(ǫ)-neighborhood. The convergence results andstability analysis rely on singular perturbation theory for non-autonomous systems. The only requirement on the set ofreference inputs involves continuous bounded derivatives, upto the second derivative for FOI-DCF and up to the thirdderivative for SOI-DCF . For the special case of dynamic inputsoffset by a static value, we show thatSOI-DCF converges to theexact dynamic average with no steady-state error. Numericalexamples show how the proposed algorithms closely track theaverage of dynamic inputs.

I. I NTRODUCTION

This paper deals with the dynamic average consensus prob-lem for a network of agents. This problem involves designingan algorithm for each agent which tracks the average ofthe network agents’ time-varying inputs using only localand neighboring agents’ information. In recent years, thedynamic average consensus problem of multi-agent systemshas attracted increasing attention due to its broad applicationin areas such as multi-robot coordination [1], distributedesti-mation [2], sensor fusion [3], [4], and distributed tracking [5].

The work [6] proposes a dynamic average consensus al-gorithm which is able to track the average of ramp refer-ence inputs with zero steady-state error. In addition to thelimited inputs that it can track, the filter is not robust toestimator initialization errors. Instead, [3] proposes a low-pass consensus filter which tracks, with a steady state error,the average of identical inputs with a uniformly boundedrate. The work [7] also proposes a consensus filter whichachieves dynamic average consensus over a common time-varying reference signals. However, the algorithm assumesthat agents know the nonlinear model which generates thetime-varying reference signals. Using input-to-state stabil-ity analysis, [8] proposes a proportional dynamic averageconsensus algorithm that can track the average of boundedreference inputs with bounded derivatives with boundedsteady-state error. This approach is generalized in [9] toachieve robust dynamic average consensus of a broad classof time-varying inputs, assuming model information aboutthem is available when designing the filter.

*This work was supported by L3 Communications through the UCSDCymer Center for Control Systems and Dynamics.

The authors are with the Department of Mechanical and AerospaceEngineering, University of California San Diego, La Jolla,CA 92093, USAskia,cortes,soniamd@ucsd.edu.

The stability and performance of the aforementioned al-gorithms are studied in a continuous-time setting. Thework [10] develops instead a discrete-time dynamic aver-age consensus estimator. The convergence analysis for theproposed dynamic average consensus algorithms relies uponthe input-to-output stability property of discrete-time staticaverage consensus algorithms in the presence of externaldisturbances. With a proper initialization of the internalstates, the proposed estimator can track, with bounded steadystate error, the average of the time-varying inputs whosenth-order difference is bounded. In the special case where thenth-order difference is asymptotically zero, the estimatesofthe average converge to the true average asymptotically withone time step delay. The conditions on the initialization,similarly to [6], makes the results in [10] not robust toinitialization errors.

In this paper, we propose two novel continuous-time dynamicaverage consensus algorithms that allow a group of agentsto track the average of their reference inputs within aO(ǫ)-neighborhood. We term these algorithms1st-Order-InputDynamic Consensus Filter(FOI-DCF) and 2nd-Order-InputDynamic Consensus Filter(SOI-DCF), respectively. Thesefilters perform the task starting from any finite initial condi-tion when the interaction network is described by a balancedand weakly-connected directed graph. The convergence andstability analysis relies on an original application to dynamicconsensus problems of singular perturbation theory of non-autonomous systems. The only requirement on the set ofreference inputs involves continuous and bounded derivativesof the inputs (up to second-order derivatives forFOI-DCFand up to third-order derivatives forSOI-DCF). For thespecial case of dynamic inputs offset by a static value,we show thatSOI-DCF converges to the exact dynamicaverage with no steady-state error. For static inputs, bothfilters converge to the exact input average. Numerical ex-amples show good tracking performance of the algorithms.Simulations also show that the proposed filters are robust tosporadic switching network topologies and tracking inputsthat are not necessarily differentiable. This is a reasonableexpected behavior if dwelling times are sufficiently large.Additionally, the proposed filters are able to handle, withno extra adjustments, permanent changes in the networktopology due to agents joining or leaving the network. Onecan attribute this robustness properties to the global stabilityand convergence properties of the proposed filters.

The paper organization is as follows. Section II introducesthe main notation used throughout the paper and reviews thestatic consensus filter proposed in [8]. Section III motivates

the use of singular perturbation theory to generate dynamicconsensus algorithms. Section IV contains the main resultson the proposed dynamic average consensus algorithms.To demonstrate the effectiveness of the proposed filters,Section V presents various numerical examples. For theconvenience of the reader, the Appendix gives a brief reviewof the Singular Perturbation problem definition and the mainresult we use to characterize the properties of our dynamicconsensus filters.

II. N OTATION AND PRELIMINARIES

The vector 1n represents ann-dimensional vector withall elements equal to one, andIn represents the identitymatrix with dimensionn. We denote byAT the transposeof matrix A. For matricesA ∈ R

n×m and B ∈ Rp×q,

we let A⊗B denote their Kronecker product. We denoteδ1(ǫ) ∈ O(δ2(ǫ)) if there exist positive constantsc and ksuch that

|δ1(ǫ)| ≤ k|δ2(ǫ)|, ∀ |ǫ| < c.

For time-varying variables, e.g.v(t), most of the time, wedrop the time dependency; only when the emphasis on thetime dependency is necessary we will use the completenotation.

In the following, we present some basic notions from al-gebraic graph theory (for more details see [11] or [12]). Adirected graph, or simply digraph, is a pairG = (V, E),whereV = 1, · · · , N is a finite set called the vertex set andE ⊆ V×V is the edge set. A graph is undirected if(u, v) ∈ Eanytime(v, u) ∈ E . We denote byA ∈ R

N×N theadjacencymatrix ofG, with the property thataij > 0 if (vi, vj) ∈ E andaij = 0, otherwise. Theout-degreeand in-degreeof a vertexvi, i ∈ 1, · · · , N, are respectively,dout(vi) =

∑n

j=1 aij

and din(vi) =∑N

j=1 aji. The out-degree matrixDout isthe diagonal matrix defined by(Dout)ii = dout(vi), for alli ∈ 1, · · · , N. The Laplacian matrix is L = Dout − A.Note thatL1N = 0. An undirected graph is calledconnectedif, for every pair of vertices inV, there is a path that has themas its end vertices. The digraph is calledweakly connectedif it is connected when viewed as an undirected graph, thatis, a disoriented digraph. Here, a digraph is calledbalancedif, for every vertex, the in-degree and out-degree are equal.The network considered here is composed ofN > 1 agentswhose communication topology is described by a weakly-connected and balanced digraph. We useD to refer to sucha graph. For this type of networks, we haverank(L) = N−1,1TNL = 0, andL+ LT is a positive semi-definite matrix.

For network problems involving internal states of dimensionn > 1, we defineLn = L⊗ In. The local variables at eachagent are distinguished by a superscripti, e.g.,ui(t) is thelocal dynamic input of agenti. If pi ∈ R

n is a local vectorat agenti, the aggregatedpi’s of the network is representedby pT =

[

p1 T · · · pN T]T∈ R

nN .

We use the following lemma from the literature to developour main results:

Lemma 2.1 (Static average consensus filter [8]):Given anetworked system with topologyD, and constant inputsui ∈ R

n, for i = 1, · · · , N , consider the following solver ateach agenti:

zi = −(zi − ui)−N∑

j=1

Lij(zj + νj),

νi =

N∑

j=1

Lijzj .

From any initial conditionz(0) ∈ Rn and ν(0) ∈ R

n, zi

converges to1N

∑N

i=1 ui exponentially.

III. M OTIVATION TO USE SINGULAR PERTURBATION

THEORY TO GENERATE CONSENSUSALGORITHMS

In this section, we motivate the use of singular perturbationtheory in devising distributed dynamic average consensusalgorithms. The simplest dynamics that generatesxi(t) →1N

∑N

i=1 ui(t), asymptotically, in each agent is the following:

xi = −(xi −1

N

N∑

i=1

ui(t)) +1

N

N∑

i=1

ui(t).

To decentralize this dynamics, we can make use of a mech-anism that generates the average of the inputs and also theaverage of the derivative of inputs,rapidly, in each agent in adistributed fashion. Then, the distributed dynamic consensusalgorithm becomes a two-time scale operation, a fast dynam-ics to generate each average and a slow dynamics to track theinput average. A singularly perturbed dynamical system isan appropriate platform to construct such an algorithm. Theaforementioned fast and slow dynamics could be realizedby means of the following—seemingly more straightforwardand systematic—mixed discrete/continuous-time algorithmrunning synchronously at each nodei ∈ 1, · · · , N:

1: (Initialization) atk = 0 initialize xi(0) ∈ Rn

2: while data existsdo3: Obtain inputsui(k) and ui(k)4: Initialize zi(0),νi(0) ∈ R

n

5: Solve the following dynamical equation, until itreaches its equilibrium point

ǫ zi(t) = −(zi(t) + ui(k) + ui(k))

−∑N

i=j Lij(zj(t) + νj(t)),

ǫ νi(t) =∑N

j=1 Lijzj(t),

(1)

6: Let zi converge to equilibriumzi

7: Define:

xi(k + 1) = xi(k)−∆t(

xi(k) + zi(k))

(2)

8: k ← k + 19: end while

In the above algorithm∆t is the time-step and0 < ǫ ≪ 1is a scalar value. Note that using the result of Lemma 2.1,

at each time stepk, the dynamical system (1) acts as astatic consensus filter with static inputui(k) + ui(k). Thisfilter converges exponentially tozi(k) = − 1

N

∑N

i=1(ui(k)+

ui(k)). For very smallǫ, the convergence rate of (1) is high.Therefore, at any time-stepk, (2) becomes:

xi(k + 1) = xi(k)−∆t

(

xi(k)−1

N

N∑

i=1

(ui(k) + ui(k))

)

,

for i = 1, · · · , N . For small ∆t, the stability and con-vergence of the above difference equation can be studiedusing the following continuous-time model (represented incompact form):

yT = −yT , (3)

where yT = xT − 1N ⊗ ( 1N

∑N

i=1 ui). The dynamical

system (3) is a stable linear system with all eigenvalues equalto −1. Therefore, (3) converges to zero exponentially. As aresult, xi in (2) converges to1

N

∑N

i=1 ui(t) exponentially,

for all i = 1, · · · , N .

The aforementioned algorithm performs dynamic averageconsensus with an exponential convergence rate. However,this algorithm is a conceptual algorithm; the cost of solv-ing (1) at each time-step is the main drawback of this algo-rithm which virtually makes it un-implementable. Inspiredby the multi time-scale structure observed above, we nextmake use of singular perturbation theory to weave togethersteps 5–7 and devise a continuous-time dynamic averageconsensus filter. By doing so, we avoid solving the fastdynamical system at each iteration of the algorithm, i.e., theslow dynamics does not need to wait for the fast dynamicsto converge.

IV. DYNAMIC CONTINUOUS-TIME CONSENSUSFILTERS

VIA SINGULARLY PERTURBEDDYNAMICS

Here, we employ singular perturbation theory to constructnew continuous-time dynamic average consensus algorithms.The main advantage of these filters is their convergenceirrespectively of the initial condition and, therefore, theirrobustness against initialization errors across the network.Furthermore, the convergence of the filters does not requireany knowledge of the dynamics generating the inputs.

Consider the following distributed filters, listed based onthecomplexity of the structure:

• 1st-Order-Input Dynamic Consensus Filter(FOI-DCF):

for i = 1, · · · , N

ǫ zi = −(zi + β ui + ui)−∑N

i=j Lij(zj + νj),

ǫ νi =∑N

j=1 Lijzj ,

(4a)

xi = −β xi − zi, (4b)

whereβ is a positive scalar.

• 2nd-Order-Input Dynamic Consensus Filter(SOI-DCF):

for i = 1, · · · , N

ǫ zi = −(zi + β ui + ui)−∑N

j=1 Lij(zj + νj)

− ǫ(β ui + ui),

ǫ νi =∑N

j=1 Lijzj ,

(5a)

xi = −β xi − zi, (5b)

whereβ is a positive scalar.

In the following, we show that these filters generate anO(ǫ)-approximation of the average of the dynamic inputs1N

∑N

i=1 ui(t), in a distributed fashion, for networks with

graph topologyD.

Theorem 4.1 (Convergence ofFOI-DCF): Consider a net-worked system with topologyD. If the first and the secondderivatives of the input signalui at each agent are continuousand bounded fort ≥ 0, then there is a small enoughǫ⋆ > 0such that, for allǫ ∈ (0, ǫ⋆], the trajectories of the filterFOI-DCF, starting from any finite initial conditions, satisfy‖xi(t)− 1

N

∑N

i=1 ui(t)‖ < O(ǫ) in finite time.

Theorem 4.2 (Convergence ofSOI-DCF): Consider a net-worked system with topologyD. If the first, the second,and the third derivatives of the input signalui at each agentare continuous and bounded fort ≥ 0, then there is a smallenoughǫ⋆ > 0 such that, for allǫ ∈ (0, ǫ⋆], the trajectories ofthe filterSOI-DCF, starting from any finite initial conditions,satisfy‖xi(t)− 1

N

∑N

i=1 ui(t)‖ < O(ǫ) in finite time.

The proof of Theorems 4.1 and 4.2 are very similar, there-fore, we present them together.

Proof of Theorems 4.1 and 4.2:The proof is basedon showing the filters satisfy the conditions of Theorem A.1globally. The boundary layer dynamics (fast dynamics) forboth FOI-DCF andSOI-DCF is (for i = 1, · · · , N )

d zi

dτ= −(zi + β ui(t) + ui(t))−

∑N

i=1 Lij(zj + νj),

d νi

dτ=∑N

i=1 Lijzj .

(6)Invoking Lemma 2.1, this fast dynamics converges to thefollowing values for each filter exponentially and globally:

zi = −1

N

N∑

i=1

(β ui + ui), i = 1, · · · , N. (7)

Substituting (7) into (4b) (similarly, in (5b)), we obtain thefollowing reduced system (slow dynamics), for the filtersFOI-DCF andSOI-DCF:

xi = −β xi +1

N

N∑

i=1

(β ui + ui), i = 1, · · · , N. (8)

Consider the following change of variables:

yi = xi −1

N

N∑

i=1

ui, i = 1, · · · , N.

In the new coordinates, (8) is represented as

yi = −β yi, i = 1, · · · , N. (9)

For β > 0, the system (9) is a stable linear system withsystem matrix eigenvalues equal to−β. Thus, (9) convergesglobally exponentially to zero, which is equivalent toxi

exponentially converging to1N

∑N

i=1 ui(t), i = 1, · · · , N .

Both filters, based on the corresponding required condi-tions for input signals, satisfy the differentiability andLip-schitz conditions of Theorem A.1 on any compact set of(xT , zT ,νT ). Thus, all the conditions of Theorem A.1 aresatisfied globally, and the estimates of (17), (18) and (19)hold, for all t ≥ 0 and for any bounded initial states.Note that the slow dynamics at each agent is globally expo-nentially approaching the average1

N

∑N

i=1 ui(t), therefore,

‖xT (t)− 1N ⊗1N

∑N

i=1 ui(t)‖ < O(ǫ) in a finite time, and

as a result‖xi(t)− 1N

∑N

i=1 ui(t)‖ < O(ǫ) in finite time.

The slow and fast dynamical analysis of the filtersFOI-DCF and SOI-DCF is exactly the same. The guaranteedconvergence bound is also of the same order. In the follow-ing, we show that the filterSOI-DCF has advantages overFOI-DCF, at the price of a slight increase in the complexityof the filter, and the extra condition on the input signals.For example, one can show that forSOI-DCF, we have∑N

i=1 xi(t) →

∑N

i=1 ui(t) as t → ∞, for any ǫ > 0.

The main advantage of the filterSOI-DCF overFOI-DCF isstated in the following result.

Corollary 4.1 (SOI-DCF for inputs offset by a static value):Consider a networked system with topologyD as inTheorem 4.1, subject to similar assumptions on the inputsignals of agents. When the difference in the input signalsis a static offset, i.e.,ui(t) = u(t) + ui where ui isa constant vector, the filterSOI-DCF converges to theexact input average with no steady-state error. That is,xi(t)→ 1

N

∑N

i=1 ui(t), for any ǫ > 0.

Proof: Consider the following change of the variables:

pi = zi + β u+ u,

qi = xi − u,, i = 1, · · · , N.

Then, we can re-write (5), as follows (compact form)

ǫ pT = −(pT + β uT )−Ln(pT + νT ),

ǫ νi(t) = LnpT ,(10a)

qT = −β qT − pT . (10b)

We can show that the equilibrium points of this systemare at (pi = − β

N

∑N

i=1 ui, qi = 1

N

∑N

i=1 ui, νi) for

i = 1, · · · , N , where

−LnνT = (pT + β uT ).

Consider the following Lyapunov function, whereqT =qT − qT , pT = pT − pT , and νT = νT − νT :

V =β

2q2T +

ǫ

8p2T +

ǫ

8ν2T .

The derivative of this Lyapunov function along the trajecto-ries of (10) is

V =−1

4pTT (Ln +L

Tn )pT − (

1

2pT − β qT )

2,

which is negative semi-definite. It is zero in the setS =pT , qT , νT ∈ R

nN | pT = 1N ⊗α, pT = 2β qT , whereα ∈ R

n. We can show thatqT = 0, qT = 0, νT =1N ⊗ γ, where γ ∈ R

n, is the smallest invariant setcontained inS. From the LaSalle Invariance Principle itnow follows that qT → 0 as t → ∞. This results inxi(t) → u(t) + 1

N

∑N

i=1 ui = 1

N

∑N

i=1 ui(t) as t → ∞,

for i = 1, · · · , N , in the filter SOI-DCF.

Remark 4.1 (Convergence for static inputs):Following anargument similar to the proof of Corollary 4.1, we can showthat, for static inputs, for anyǫ > 0, both proposed filtersconverge to the exact input average with no steady-state error.

Remark 4.2 (Role ofβ): As it is shown in the proof ofTheorems 4.1, and 4.2,β is the rate of convergence of theslow dynamics. By choosing a largeβ, we can increasethe rate of convergence. However, to keep the two time-scale structure of the filters,β has to be chosen smallerrelative to ǫ−1. Quantifying the convergence neighborhoodand the effect ofβ on the size of this neighborhood is leftas a future work. Furthermore, one should notice thatβ isa global variable known to all agents in the network. Toguarantee convergence to the right dynamic input averageof 1

N

∑N

i=1 ui(t), in its O(ǫ) neighborhood, every agent

should agree upon a common value forβ before running theconsensus filters. The following filter allows the agents to usedifferent values forβ and still converge to the right dynamicaverage. However, note that to provide this robustness withrespect toβ, we are requiring extra communication channels.

for i = 1, · · · , N, (11a)

ǫ zi = −(zi + ui)−∑N

i=j Lij(zj + νj),

ǫ νi =∑N

j=1 Lijzj ,

(11b)

ǫ yi = −(yi + ui)−∑N

i=j Lij(yj + µj),

ǫ µi =∑N

j=1 Lijyj ,

(11c)

xi = −βi xi − βi zi − yi, (11d)

whereβi’s are all positive scalars. To guarantee convergenceto the O(ǫ) neighborhood of the dynamic input average,the requirement on the input signals is the boundednessand continuity of their first and second derivatives. Noticethat using differentβi results in different convergence ratesat each agent. Then one can expect that the tracking isnot coherent across the network agents. The stability andconvergence analysis of this filter is along the same lines ofthe proof of Theorem 4.1 and omitted for brevity.

Fig. 1. Network

V. NUMERICAL EXAMPLES

In order to give the reviewers a demonstration of the differentaspects of the proposed filters, in the following we provide anextensive set of numerical examples. However, in the finalversion of the paper, due to the space limitation, we needto reduce the number of examples. Here, we mainly performsimulations using the filterFOI-DCF. A more comprehensiveset of examples including simulations conducted using thefilter SOI-DCF can be found in [13].

In the following simulations, the thick dashed line is thedynamic input average and the thin colored lines are thetime histories of the localxi states of the filters. In all thesimulations below, all the initial conditions for the proposedfilters here are selected uniformly randomly inU [−2, 2].

A. Example 1 (Performance evaluation of the filterFOI-DCF over a large network)

Consider the randomly generated undirected network (usingMatlab BGL package [14]) shown in Fig. 1 which consistsof N = 100 agents. The local input signals are

ui(t) = ai sin(bi t+ ci), i = 1, · · · , N,

where the input coefficients are generated randomly uni-formly in the following ranges:ai ∼ U [−5, 5], bi ∼ U [1, 2],ci ∼ U [0, π/2].

Figures 2 shows the time histories of the local internal statesxi generated by the filterFOI-DCF and the dynamic inputaverage for the three cases of (ǫ = 0.01, β = 1), (ǫ = 0.001,β = 1) and (ǫ = 0.001, β = 3). As expected, the smaller theǫ is, the better the tracking, and the larger theβ, the fasterthe convergence to theO(ǫ) neighborhood is.

B. Example 2 (Comparison between the filterFOI-DCF andthe FODAC algorithm of [10])

Consider the weakly-connected and balanced directed graphin Fig. 3 . The input signals are

u1(t) = 5 sin(t) + 10t+2 + 1, u2(t) = 5 sin(t) + 10

(t+2)2 + 1,

u3(t) = 5 sin(t) + 10(t+2)3 + 1,

u4(t) = 5 sin(t) + 10 e−t + 4, u5(t) = 5 sin(t).

0 2 4 6 8 10 12 14 16 18 20−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

time

xi

(a) ǫ = 0.01 andβ = 1

0 2 4 6 8 10 12 14 16 18 20−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

time

xi

(b) ǫ = 0.001 andβ = 1

0 2 4 6 8 10 12 14 16 18 20−2

−1.5

−1

−0.5

0

0.5

1

1.5

2

time

xi(c) ǫ = 0.001 andβ = 3

Fig. 2. Simulation results for example 1

1

2

3

4

5

Fig. 3. Network of example 2

In this example, we compare the performance of the pro-posed filterFOI-DCF with the First-Order Dynamic AverageConsensus (FODAC) algorithm of [10]. Figures 4 shows theresult of the simulations. To generate the simulations forFODAC algorithm of [10], we used the step sizeh = 0.001which is higher than the bound for guaranteed convergence(to accelerate the simulations). As Fig. 4 shows, the FO-DAC algorithm of [10] starting from the required initialconditions ofxi(0) = ui(−h), i = 1, · · · , N , has a higherrate of convergence than the filterFOI-DCF. However, asdemonstrated in Fig. 3(c), for initial conditions other thanxi(0) = ui(−h) this filter produces a large steady sateerror. In this simulation, for the filterFOI-DCF, we usedǫ = 0.01 and β = 1. In case of agent failure, the FODACalgorithm requires an adjustment of the initial condition inorder to guarantee tracking. As we see in the following thisadjustment is necessary in our simulated algorithms.

0 1 2 3 4 5 6 7 8 9 10−4

−2

0

2

4

6

8

time

xi

(a) Filter FOI-DCF with ǫ = 0.01 andβ = 1

0 1 2 3 4 5 6 7 8 9 10−5

0

5

10

15

time

xi

0 0.01 0.02

0

5

10

15

(b) The FODAC algorithm of [10] withxi(0) = ui(−h)

0 1 2 3 4 5 6 7 8 9 10−10

−8

−6

−4

−2

0

2

4

6

8

time

xi

(c) The FODAC algorithm of [10] withxi(0) ∈ U [−2, 2]

Fig. 4. Simulation results for example 2.

C. Example 3 (Performance of the filterFOI-DCF over anetwork with time-varying topology)

Consider the time-varying weakly-connected and balanceddirected graph in Fig. 5. The input signals are

u1(t) = 6 cos(0.2 t), u2(t) = atan(0.2 t),

u3(t) = 3 sin(0.1 t+ 1), u4(t) = log(t+ 0.1),

u5(t) = 0.02 t, u6(t) = 0.5 (t− 25).

The simulation result is shown in Fig. 6. The onset of anychanges in the network topology is indicated by verticaldotted lines. In Fig. 6, in order to convey clearly theincidences of the agents joining or leaving the network, forthe time prior to joining and the time after departure, thefixed initial value or departure value is used, respectively.As the simulation shows in Fig. 6, the proposed filterFOI-DCF handles the changes in the topology well. When thechange is just a switching in the network communicationthe filter is almost indifferent to the change. In the casesof agents joining or leaving the network, after a transientperiod, the filter presumes its close tracking of the inputaverage. Here, we usedβ = 1 and ǫ = 0.01.

D. Example 4 (Comparison between the performance of theproposed filtersFOI-DCF and SOI-DCFwhen input signalsare offset with a static value)

Consider the network depicted in Fig. 3 with the input signalslisted below:

u1(t) = u(t) + 1, u2(t) = u(t)− 1, u3(t) = u(t) + 4,

u4(t) = u(t) + 5, u5(t) = u(t) + 10,

2

3

4

5

6

2

3

4

5

t ∈ [40, ∞)

1

2

3

4

5

t ∈ [10, 15)

2

3

4

5

2

3

4

5

6

t ∈ [15, 25) t ∈ [25, 30)

t ∈ [30, 40)

1

2

3

4

5

t ∈ [0, 10)

Fig. 5. Network of example 3

0 10 20 30 40 50 60−3

−2

−1

0

1

2

3

time

xi

x1

x2

x3

x4

x5

x6

input average

Fig. 6. Simulation result for example 3.

where u(t) = 5 sin(t). These inputs differ from one andother by a static value. Figure 7 shows the time history of theerror between the statesxi’s and the dynamic input average.The dashed blue lines belong to the filterFOI-DCF and thesolid red lines belong toSOI-DCF. Figure 7(a) is generatedusing ǫ = 1, and Fig 7(b) is generated usingǫ = 0.01.As expected the filterSOI-DCF, regardless of the value ofǫ, converges to the dynamic input average with no steady-state error. For a smallǫ, the performance of the filterFOI-DCF improves and almost matches the perfect performanceof the filter SOI-DCF.

E. Example 5 (Comparison between the performance of theproposed filtersFOI-DCF and (11) when agents do not usecommonβ)

Consider the network depicted in Fig. 3 with the input signalslisted in Example 2. Here, we compare the performance ofthe filtersFOI-DCF and (11), when agents use the followingβi’s

β1 = 1.2, β2 = 1, β3 = 0.8, β4 = 0.5, β5 = 0.2.

0 5 10 15 20 25 30−5

0

5

10

time

error

(a) ǫ = 1 andβ = 1

0 5 10 15 20 25 30−1

0

2

4

6

time

error

(b) ǫ = 0.01 andβ = 1

Fig. 7. Simulation results for example 4.

0 2 4 6 8 10 12 14 16 18 20−4

−2

0

2

4

6

8

10

12

time

xi

(a) The proposed filtersFOI-DCF

0 2 4 6 8 10 12 14 16 18 20−6

−4

−2

0

2

4

6

8

time

xi

x1

x2

x3

x4

x5

input average

(b) the proposed filter (11)

Fig. 8. Simulation results for example 5.

Simulation result, depicted in Fig. 8(a), indicate that thefilter FOI-DCF does not approach to the right dynamic inputaverage, as it is not robust with respect toβ. However, thefilter (11), which is designed to work for different valuesof β at each agent, converges to the input average. For thefilter (11), as expected, the convergence rate at each agent isdefined by its correspondingβi; the larger theβi; the fasterthe convergence is. Here we usedǫ = 0.001.

F. Example 6 (performance of the proposed filtersFOI-DCF with respect to discontinuous inputs)

Consider the network depicted in Fig. 3. The followings arethe inputs at each agent

u1(t) = 3u(t) cos(0.2 t), u2(t) = u(t) tanh(0.2 t),

u3(t) = 3u(t) sin(0.1 t+ 1), u4(t) = u(t) log(t+ 0.1),

u5(t) = 0.02u(t) t.

whereu(t) =∑

∞

i=0((−1)iH(t − 20 i)), in which H is the

step function, i.e.,

H(t) =

0 t < 0,

1 t ≥ 0.

0 50 100 150−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

3

time

Fig. 9. Simulation result for example 6.

The simulation result obtained using the proposed filterFOI-DCF is shown in Fig. 9. Here we usedǫ = 0.01. Asshown in the figure, the proposed filterFOI-DCF achievesa close tracking despite the discontinuity in the input signalsas long as dwell times are sufficiently large.

VI. CONCLUSIONS

We have proposed two continuous-time dynamic averageconsensus filters for networks with balanced and weakly-connected directed graph topology. The proposed filters havea two-time scale structure and do not require model infor-mation on the dynamic inputs. Using singular perturbationanalysis, we have shown that the proposed filters reach anO(ǫ)-neighborhood of the dynamic input average in finitetime irrespectively of the initial condition. Simulation resultsshow that the filters are robust to switching network topolo-gies, as long as the network stays balanced and weakly-connected, and there is a large enough dwelling time betweeneach switchings. Future work will be devoted to rigorouslycharacterizing the convergence properties of the proposeddynamic average consensus algorithms over networks withswitching topology.

REFERENCES

[1] P. Yang, R. A. Freeman, and K. M. Lynch, “Mutli-agent coordina-tion by decentralized estimation and control,”IEEE Transactions onAutomatic Control, vol. 53, no. 11, pp. 2480–2496,, 2008.

[2] S. Meyn, Control Techniques for Complex Networks. CambridgeUniversity Press, 2007.

[3] R. Olfati-Saber and J. S. Shamma, “Consensus filters for sensornetworks and distributed sensor fusion,” inIEEE Conf. on Decisionand Control and European Control Conference, (Seville, Spain),pp. 6698–6703, December 2005.

[4] R. Olfati-Saber, “Distributed Kalman filtering for sensor networks,” inIEEE Conf. on Decision and Control, (New Orleans, LA), pp. 5492–5498, Dec. 2007.

[5] P. Yang, R. A. Freeman, and K. M. Lynch, “Distributed cooperativeactive sensing using consensus filters,” inIEEE International Confer-ence on Robotics and Automation, (Roma, Italy), pp. 405–410, April2007.

[6] D. P. Spanos, R. Olfati-Saber, and R. M. Murray, “Approximatedistributed Kalman filtering in sensor networks with quantifiableperformance,” inSymposium on Information Processing of SensorNetworks, (Los Angeles, CA), pp. 133–139, Apr. 2005.

[7] W. Ren, “Consensus seeking in multi-vehicle systems with atime-varying reference state,” inAmerican Control Conference, (New York,USA), pp. 717–722, July 2007.

[8] R. A. Freeman, P. Yang, and K. M. Lynch, “Stability and conver-gence properties of dynamic average consensus estimators,” in 2006Conference on Decision and Control, (San Diego, CA), pp. 398–403,2006.

[9] H. Bai, R. A. Freeman, and K. M. Lynch, “Robust dynamic averageconsensus of time-varying inputs,” inIEEE Conference on Decisionand Control, (Atlanta, GA, USA), pp. 3104–3109, December 2010.

[10] M. Zhu and S. Martınez, “Discrete-time dynamic average consensus,”Automatica, vol. 46, no. 2, pp. 322–329, 2010.

[11] F. Bullo, J. Cortes, and S. Martınez, Distributed Control of RoboticNetworks. Applied Mathematics Series, Princeton University Press,2009. Available at http://www.coordinationbook.info.

[12] M. Mesbahi and M. Egerstedt,Graph theoretic methods in multia-gent networks. Princeton Series in Applied Mathematics, PrincetonUniversity Press, 2010.

[13] S. S. Kia, J. Cortes, and S. Martınez, “Singularly perturbed filters fordynamic average consensus (extended version),” Oct. 2012. Availableat tintoretto.ucsd.edu/solmaz/Papers/cp17.html.

[14] D. F. Gleich,Models and Algorithms for PageRank Sensitivity. PhDthesis, Stanford University, September 2009. Chapter 7 on Matlab-BGL.

[15] H. K. Khalil, Nonlinear Systems. Prentice Hall, 3 ed., 2002.

APPENDIX

For the sake of completeness, here we give a short reviewof Singularly Perturbed dynamical system theory and itsterminology along with the theorem that we will employto derive our main result on dynamic average consensusalgorithms (see [15] for more details).

Consider the following system:

x = f(t,x, z, ǫ), x(t0) = η(ǫ), (12a)

ǫ z = g(t,x, z, ǫ), z(t0) = ζ(ǫ). (12b)

The use of a small constantǫ > 0 induces two time scalesin the system, resulting into a fast and a slow dynamics. Theanalysis of such systems can be achieved with the aid ofSingular Perturbation Theory. Singular Perturbation Theoryestablishes rigorous conditions under which the behavior ofthe system follows that of the limiting system whenǫ goesto 0. We assume thatf andg are continuously differentiablein their arguments for(t,x, z, ǫ) ∈ [0,∞) × Dx × Dz ×[0, ǫ0], whereDx ⊂ R

n andDz ⊂ Rm are open connected

sets. When we setǫ = 0 in (12), the dimension of the stateequation reduces fromn + m to n because the differentialequation (12b) degenerates into the algebraic equation

0 = g(t,x, z, 0). (13)

We say that the model (12) is in standard form if (13) hask ≥ 1 isolated real roots

zi = hi(t,x), i = 1, · · · , k, (14)

for each(t,x) ∈ [0,∞)×Dx. This assumption assures that awell-definedn−dimensionalreduced model(slow dynamics)will correspond to each root of (13). To obtain theith reducedmodel, we substitute (14) into (12a), atǫ = 0, to obtain

x = f(t,x,h(t,x), 0), (15)

where we have dropped the subscripti from h. Theboundary-layersystem (fast dynamics) is

dz

dτ= g(t,x, z, 0), τ =

t

ǫ, (16)

wherex andt are treated as fixed parameters. In the analysis,it is common to perform the change of variablesy = z −h(t,x) that shifts the quasi-steady state ofz to the origin.

Theorem A.1 ([15]):Consider the Singular Perturbationproblem of (12) and letz = h(t,x) be an isolated root of(13). Assume that, for all

[t,x, z− h(t,x), ǫ] ∈ [0,∞)×Dx ×Dy × [0, ǫ0],

whereDx ⊂ Rn andDy ⊂ R

m are domains which containtheir respective origins, the following conditions hold:

• On any compact subset ofDx × Dy, the functionsf , g, their partial derivatives with respect to(x, z, ǫ),and the first partial derivative ofg with respect tot are continuous and bounded; the functionsh(t,x)and[∂g(t,x, z, 0)/∂z] have bounded first partial deriva-tives with respect to their arguments; the function[∂f(t,x,h(t,x), 0)/∂x] is Lipschitz inx, uniformly int; and the initial conditionsη(ǫ) and ζ(ǫ) are smoothfunctions ofǫ;

• the origin is an exponentially stable equilibrium pointof the reduced system (15); that is, there is a Lyapunovfunction V (t,x) that satisfies the conditions of [15,Theorem 4.9] for (15) for(t,x) ∈ [0,∞) × Dx. Inother words, we have

W1(x) ≤ V (t,x) ≤W2(x),

∂V

∂t+

∂V

∂xf(t,x,h(t,x), 0) ≤ −W3(x),

whereW1, W2, andW3 are continuous positive definitefunctions onDx, such thatW1(x) ≤ c is a compactsubset ofDx;

• the origin is an exponentially stable equilibrium of theboundary layer system, uniformly in(t,x). Let Ry ⊂Dy be the region of attraction of

dy

dτ= g(0,η(0),y + h(0,η(0)), 0),

y(0) = ζ(0)− h(0,η(0)),

andΩy be a compact subset ofRy.

Then, for each compact setΩx ⊂ W2(x) ≤ ρc, 0 < ρ < 1there is a positive constantǫ⋆ such that for allt0 ≥ 0, η0 ∈Ωx, ζ0 − h(t0,η0) ∈ Ωy, and 0 < ǫ < ǫ⋆, the singularlyperturbed system (12) has a unique solutionx(t, ǫ), z(t, ǫ)on [t0,∞), and

x(t, ǫ)− x(t) = O(ǫ), (17)

z(t, ǫ)− h(t, x(t))− y(t/ǫ) = O(ǫ) (18)

hold uniformly for t ∈ [t0,∞), where x(t) and y(τ) arethe solutions of the reduced and boundary layer problems.Moreover, given anytb > t0, there isǫ⋆⋆ ≤ ǫ⋆ such that

z(t, ǫ)− h(t, x(t)) = O(ǫ) (19)

holds uniformly fort ∈ [tb,∞) wheneverǫ < ǫ⋆⋆.